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Properties

Label 2-370-185.184-c1-0-10
Degree $2$
Conductor $370$
Sign $0.964 - 0.264i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.987i·3-s + 4-s + (1.85 + 1.25i)5-s − 0.987i·6-s + 4.78i·7-s + 8-s + 2.02·9-s + (1.85 + 1.25i)10-s − 5.98·11-s − 0.987i·12-s + 3.49·13-s + 4.78i·14-s + (1.23 − 1.83i)15-s + 16-s − 4.96·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.570i·3-s + 0.5·4-s + (0.829 + 0.559i)5-s − 0.403i·6-s + 1.81i·7-s + 0.353·8-s + 0.674·9-s + (0.586 + 0.395i)10-s − 1.80·11-s − 0.285i·12-s + 0.969·13-s + 1.28i·14-s + (0.318 − 0.472i)15-s + 0.250·16-s − 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.964 - 0.264i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.964 - 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23117 + 0.300616i\)
\(L(\frac12)\) \(\approx\) \(2.23117 + 0.300616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-1.85 - 1.25i)T \)
37 \( 1 + (-3.96 + 4.61i)T \)
good3 \( 1 + 0.987iT - 3T^{2} \)
7 \( 1 - 4.78iT - 7T^{2} \)
11 \( 1 + 5.98T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + 7.33iT - 19T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + 3.24iT - 31T^{2} \)
41 \( 1 + 0.530T + 41T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 - 4.30iT - 47T^{2} \)
53 \( 1 - 3.66iT - 53T^{2} \)
59 \( 1 - 2.15iT - 59T^{2} \)
61 \( 1 - 3.06iT - 61T^{2} \)
67 \( 1 - 3.79iT - 67T^{2} \)
71 \( 1 + 8.47T + 71T^{2} \)
73 \( 1 + 9.05iT - 73T^{2} \)
79 \( 1 - 5.56iT - 79T^{2} \)
83 \( 1 - 3.77iT - 83T^{2} \)
89 \( 1 - 8.45iT - 89T^{2} \)
97 \( 1 + 3.64T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.43903107105458665121583980901, −10.81623396817877609723916059085, −9.608737478237245253293202191110, −8.640023195808656264034070397819, −7.47212179864919847760130083928, −6.37408445897663903710792439895, −5.75136725446618750274972646259, −4.72357465053529621543121866349, −2.66097963354595395714270403177, −2.27042460515026142505502233922, 1.53376509954211890267685766030, 3.42642293707260447109328845837, 4.45967534570628035545044518177, 5.19645578868219650530748441278, 6.45972506187380997403812078953, 7.46530806170326629495335289006, 8.524700129239304176207466938723, 10.02155883977831589609270242173, 10.40813087938178799978309383213, 11.01912038682803575015850064491

Graph of the $Z$-function along the critical line